Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,21)(16,30)(17,23)(18,24)(19,25)(20,26)(22,27)(28,38)(29,39)(31,43)(32,42)(33,41)(34,37)(35,44)(36,40);(2,4)(5,7,8,11)(6,9,10,12)(13,15,19,29)(14,21,25,39)(16,31,27,36)(17,32,34,33)(18,28,20,35)(22,40,30,43)(23,42,37,41)(24,38,26,44);(6,10)(9,12)(13,16,19,27)(14,22,25,30)(15,17,29,34)(18,33,20,32)(21,37,39,23)(24,42,26,41)(28,36,35,31)(38,43,44,40);(13,17)(14,23)(15,27)(16,29)(18,31)(19,34)(20,36)(21,22)(24,43)(25,37)(26,40)(28,33)(30,39)(32,35)(38,41)(42,44);(13,18,19,20)(14,24,25,26)(15,28,29,35)(16,32,27,33)(17,31,34,36)(21,38,39,44)(22,41,30,42)(23,43,37,40)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347728);